Related papers: Rasmussen's spectral sequences and the sl(N)-conco…
We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise from sl(n) knot cohomology. We verify some properties which…
We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N>0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an…
We construct the concordance invariant coming from the $E(-1)$ spectral sequence on Khovanov homology in the same way Rasmussen's $s$ invariant comes from the Lee spectral sequence, and show that it gives a bound on the nonorientable slice…
We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward…
We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$-invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{|L|}$. The basic…
We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.
We define a deformation of our earlier link homologies for fundamental representations of sl_m. The deformed homology of a link is isomorphic to the deformed homology of the disjoint union of its components. Moreover, there exists a…
A well-known conjecture of Rasmussen states that for any knot $K$ in $S^{3}$, the rank of the reduced Khovanov homology of $K$ is greater than or equal to the rank of the reduced knot Floer homology of $K$. This rank inequality is supposed…
In this paper we construct a sequence of integer-valued concordance invariants $\nu_n(K)$ that generalize the Ozsv\'ath-Szab\'o $\nu$-invariant and the Hom-Wu $\nu^+$-invariant.
A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2…
We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot…
Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally…
We define and study a family of link invariants $\mathit{HFK}_{n}(L)$. Although these homology theories are defined using holomorphic disc counts, they share many properties with $sl_{n}$ homology. Using these theories, we give a framework…
By means of Rasmussen's formulation of Khovanov-Rozansky homology originally given over $\mathbb{Q}$ in arXiv:math/0607544, we compare different flavors of $\mathfrak{sl}(n)$ link homology with the link invariants obtained by Kitchloo in…
We investigate the filtered theory corresponding to the universal sl(2) foam cohomology $H_{a,h}$ for links, where a and h are complex numbers. We show that there is a spectral sequence converging to $H_{a,h}$ which is invariant under the…
We establish two spectral sequences in knot Floer homology associated to a directed strongly invertible knot K: one from the knot Floer homology of K to a two dimensional vector space, and one from the singular knot Floer homology of a…
It is well-known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's concordance invariant s. This gives a concordance homomorphism to the integers and a strong lower…
This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to $\mathfrak{sl}(n)$ homologies. We extend Levine and Zemke's ribbon concordance obstruction…
In this survey article, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsvath and Szabo. Along the way, we prove that if two knots are concordant, then their knot Floer complexes…
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new…